Non-stationary MHD interaction of a horizontal magnetic field with a three-dimensional cellular convection is studied by means of computational methods and methods of mean field electrodynamics.

For a given magnetic field drop across the convective layer, the rate of magnetic flux penetration through this layer is characterized by two integral coefficients: the first one describing the topological pumping effect arises from the antisymmetric part of the α-effect, while the second coefficient accounts for the enhancement of the effective diffusion due to the convective motions. In the magnetic-Reynolds-number range studied (−5 [les ] Rm [les ] 5) these coefficients are found to be, correspondingly, odd and even functions of Rm only. The net magnetic flux escape rate into vacuum decreases at Rm > 2·2 when compared with a case of a layer without cellular motions. Here the topological pumping prevails not only over the convective enhancement of diffusion but begins to suppress even the background diffusion action.

Thus, the asymmetry in the transport properties of cellular motion is again demonstrated, and their difference from those of random turbulence is identified.

A standard approach to propulsion velocities of autophoretic colloids with thin interaction layers uses a reciprocity relation applied to the slip velocity although the surface flux (chemical, electrical, thermal, etc.), which is the source of the field driving the slip, is often more accessible. We show how, under conditions of low Reynolds number and a field obeying the Laplace equation in the outer region, the slip velocity can be bypassed in velocity calculations. In a sense, the actual slip velocity and a normal field proportional to the flux density are equivalent for this type of calculation. Using known results for surface traction induced by rotating or translating an inert particle in a quiescent fluid, we derive simple and explicit integral formulas for translational and rotational velocities of arbitrary spheroidal and slender-body autophoretic colloids.

We obtain approximate solutions to the equations that govern the shape of giant unilamellar vesicles (GUVs) with two fluid phases. The equations involve a dimensionless small parameter related to the resistance to changes in its local mean curvature. Asymptotic solutions for the shape are obtained up to and including terms of first order in the small parameter. At this order, we determine a relationship between the tangent angle at the interface and the difference in the Gaussian curvature stiffnesses of the co-existing phases. This relationship demonstrates that a difference in the Gaussian curvature stiffnesses moves the phase boundary away from the neck, as determined in previous numerical studies. The analytical expression for the tangent angle obtained here can be used to determine elastic parameters for the membranes from experimental data. Use of the analytical expression will eliminate the need for the repeated generation of numerical solutions in the estimation of the material parameters. Our analytical solution also reduces the number of measurements needed as inputs for an existing boundary layer analysis.

The coupling of the temperature and velocity fields by buoyancy in a laminar two-dimensional wall jet over a finite-length horizontal plate is studied numerically and analytically in the asymptotic limit of infinite Reynolds number. Two configurations are considered leading to a cold layer of fluid over the plate, namely an ambient-temperature jet over a cooled plate and a cold jet over an insulated plate. In both cases buoyancy generates an adverse pressure gradient that may separate the flow if the Froude number is sufficiently small and always makes the solution everywhere over the plate dependent on the conditions at the downstream boundary. In the limit of very small Froude number separation occurs in a viscous–inviscid interaction region near the origin of the jet, leading to a separation bubble that covers a fraction of the plate dependent on the Prandtl number. The scalings of the solution in this asymptotic limit are obtained by order of magnitude estimations in the different regions of the bubble and in the buoyancy-dominated flow beyond the bubble, and the results are checked against the numerical solutions of the boundary layer equations. A separate analysis is carried out for very large Prandtl numbers showing that the recirculation bubble is then much shorter than the plate, also in agreement with the numerical results.

Experimental measurements were made of the difference in pressure at large distances on either side of a spherical particle settling slowly along the axis of a long circular tube filled with viscous liquid. At low particle Reynolds numbers and for small sphere-cylinder diameter ratios, it was found that the product of the pressure difference and the cross-sectional area of the tube is equal to twice the drag on the particle, in accord with theory.

In a recent work Longuet-Higgins & Stewart (1961) have studied the changes in wavelength and amplitude of progressive waves of constant frequency as they are propagated into regions of surface divergence or convergence. In the work here described the complementary conditions are assumed. Standing waves of uniform wavelength, λ, exist in an area of uniform surface divergence. Changes in amplitude and wavelength are studied. These changes depend on the existence of the radiation stress which was discovered by Longuet-Higgins & Stewart but the physical interpretation of this stress is simpler for standing than for progressive waves. Three different ways of obtaining the same rate of strain in the direction of the current caused the amplitude to vary as $\lambda|^{-{\frac {1}{4}}}, \lambda|^{-{\frac {3}{4}}}$ and $\lambda|^{-{\frac {5}{4}}}$, respectively.

Experiments in which free-standing waves were generated in a tank one wave-length wide which was then made narrower verified the conclusion that contraction does not alter the periodic character of the waves, even though the ratio of amplitude to wavelength becomes so great that they can no longer be treated mathematically by the usual linearized approximation. The shape of the profile then appears to agree well with calculations of Penney & Price (1952).

We investigate the linear stability and nonlinear evolution of a three-dimensional toroidal vortex of uniform potential vorticity under the quasi-geostrophic approximation. The torus can undergo a primary instability leading to the formation of a circular array of vortices, whose radius is approximately the same as the major radius of the torus. This occurs for azimuthal instability mode numbers $m\geqslant 3$, on sufficiently thin tori. The number of vortices corresponds to the azimuthal mode number of the most unstable mode growing on the torus. This value of $m$ depends on the ratio of the torus’ major radius to its minor radius, with thin tori favouring high mode $m$ values. The resulting array is stable when $m=4$ and $m=5$ and unstable when $m=3$ and $m\geqslant 6$. When $m=3$ the array has barely formed before it collapses towards its centre with the ejection of filamentary debris. When $m=6$ the vortices exhibit oscillatory staggering, and when $m\geqslant 7$ they exhibit irregular staggering followed by substantial vortex migration, e.g. of one vortex to the centre when $m=7$. We also investigate the effect of an additional vortex located at the centre of the torus. This vortex alters the stability properties of the torus as well as the stability properties of the circular vortex array formed from the primary toroidal instability. We show that a like-signed central vortex may stabilise a circular $m$-vortex array with $m\geqslant 6$.

We report results of a combined numerical and experimental study on axisymmetric and non-axisymmetric flow states in a finite-length, co-rotating Taylor–Couette system in the Taylor vortex regime but also in the Rayleigh stable regime for moderate Reynolds numbers (${\leq }1000$). We found the dominant boundary-driven axisymmetric circulation to play a crucial role in the mode selection and the bifurcation behaviour in this flow. A sequence of partially hysteretic transitions to other axisymmetric multi-cell flow states is observed. Furthermore, we observed spiral states bifurcating via a supercritical Hopf bifurcation out of these multi-cell states which strongly determine the shape of the spiral. Finally, an excellent agreement between experimental and numerical results of the full Navier–Stokes equations is found.

This paper is the second of a pair describing two-point velocity measurements in fully developed pipe flow. A method of processing hot-film anemometer signals to identify intervals of high energy production (‘bursts’) in wall turbulence is presented. The method uses filtered cross-stream spatial derivatives of the axial velocity fluctuations. It is demonstrated to be more sensitive to ‘bursts’ than several other methods of indentification. The bursts identified in this manner are shown to have similar characteristics to those observed in visual studies.

The technique has been applied to the wall region of turbulent pipe flow. Mean burst rates have been obtained at various distances from the wall for three Reynolds numbers. It is shown that the mean burst rate cannot be reliably obtained from a previously used technique based on the autocorrelation of the axial velocity fluctuations.

On the basis of our experiments, the mean burst rate and the turbulent shear stress have been found to vary similarly with distance from the wall. In the region near the wall where the shear stress is constant the mean burst rate is independent of the kinematic viscosity.

Some characteristics of the velocity fluctuations during burst intervals have been studied. All the bursts began with a relative minimum in the axial velocity fluctuations followed by a peak in the cross-stream spatial derivative. A second peak always occurred midway through the burst. The sequence of events is somewhat similar to that in the last stage of laminar-to-turbulent transition.

An analysis is made of the small-amplitude capillary–gravity waves which occur on the interface of two fluids and which arise out of the interaction between the Mth and Nth harmonics of the fundamental mode. The method employed is that of multiple scales in both space and time and a pair of coupled nonlinear partial differential equations for the slowly varying wave amplitudes is derived. These equations describe, correct up to third order, the progression of a wavetrain and are generalizations of the nonlinear Schrödinger-type equations used by many authors to model wave propagation. The equations are solved and formal power series expansions of the corresponding wave profiles obtained. Many different wave configurations can arise, some symmetric others asymmetric. It is found that an important influence on the type of waves which can occur is whether the ratio of the interacting wave modes is greater or less than two. Finally, an examination of the stability of the waves to plane wave perturbations is carried out.

A weak expansion wave propagating in a relaxing gas is discussed with particular reference to the ‘near-equilibrium’ and ‘near-frozen’ regions. The concept of bulk viscosity is used in conjunction with Burger's equation in the near-equilibrium region. The asymptotic equilibrium simple wave is modified by diffusive regions in the neighbourhood of the first and last rays. It is shown that in the case of a weak expansion wave, Chu's asymptotic solution of the acoustic equation describes the wave-form for a finite time interval before convection effects become noticeable. In the near-frozen region a characteristic perturbation method is used to describe the flow near the wave-front.

We show experimentally that the mechanisms of energy transport in turbulent flows of superfluid $^{4}\text{He}$ are strikingly different from those occurring in turbulent flows of viscous fluids. We argue that the result can be related to the role played by quantized vortices in this unique type of turbulence. The flow-induced motions of relatively small particles suspended in the liquid reveal that, for scales of the order of the mean distance between the vortices, the particles do not tend on average to decelerate faster than they accelerate, whereas, at larger scales, a classical-like asymmetry is recovered. It follows that, in the range of investigated parameters, flight-crash events are less apparent than in classical turbulence. We specifically link the outcome to the time symmetry of quantized vortex reconnections observed at scales comparable to the typical particle size.

In 1880, Stokes famously demonstrated that the singularity that occurs at the crest of the steepest possible water wave in infinite depth must correspond to a corner of $120^{\circ }$. Here, the complex velocity scales like $f^{1/3}$ where $f$ is the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity $f=f^{\ast }$ moves into the complex plane, and is of order $(f-f^{\ast })^{1/2}$ (Grant J. Fluid Mech., vol. 59, 1973, pp. 257–262). Grant conjectured that as the highest wave is approached, other singularities must coalesce at the crest so as to cancel the square-root behaviour. Despite recent advances, the complete singularity structure of the Stokes wave is still not well understood. In this work, we develop numerical methods for constructing the Riemann surface that represents the extension of the water wave into the complex plane. We show that a countably infinite number of distinct singularities exist on other branches of the solution, and that these singularities coalesce as Stokes’ highest wave is approached.

We examine the stability of laminar high-Reynolds-number flow through an asymmetric flexible-walled channel driven by a fixed upstream flux and subject to a (large) uniform external pressure. We construct a long-wavelength, spatially one-dimensional model using a flow profile assumption, modelling the flexible wall as a thin tensioned membrane subject to a large axial pre-stress. We numerically construct the non-uniform static shape of the flexible wall and consider its stability using both a global eigensolver and numerical simulation of the nonlinear governing equations. The system admits multiple static solutions, including a highly collapsed steady state where the membrane has a single constriction which increases with increasing external pressure. We demonstrate that the non-uniform static state is unstable to two distinct (infinite) families of normal modes which we characterise in the limit of large external pressure. In particular, there is a family of low-frequency oscillatory modes which each persist to low membrane tensions, where the most unstable mode has an oscillating membrane profile which is outwardly bulged at the centre of the domain with a narrow constriction at the downstream end. In addition, there is a family of high-frequency oscillatory modes which are each unstable beyond a critical value of the tension within a two-branch neutral curve. Unstable modes along the lower branch of the neutral curve are sustained by a leading-order balance between unsteady inertia and the restoring force of membrane tension along the channel. In addition, we elucidate the mechanism of energy transfer to sustain the self-excited oscillations: oscillations decrease the mean maximal constriction of the channel over a period, which reduces the overall dissipation of the mean flow and releases energy to sustain the instability. Fully nonlinear simulations indicate that as the Reynolds number increases these unstable normal modes can grow supercritically into sustained large-amplitude ‘slamming’ oscillations, where the membrane is periodically drawn very close to the opposite rigid wall before recovering.